Preface: I did not invent this build, nor do I claim to be the world's expert at it, however I play this build, and noticed that a consolidated source of information about it was lacking on these forums, so I took it upon myself to assemble what information I could about the build in one place, for your convenience.
How it works:
The goal of this build is to generate sick amounts of fury, so you can spam sprint and whirlwind. Sprint - Run Like the Wind is actually your primary damage dealer, as well as your primary fury generator, and your primary source of LoH healing. Everything else about the build is designed either to optimize your ability to keep sprint active, or your ability to move around and drop sprint tornadoes. People call this the whirlwind build, but whirlwind is just a tool for letting you move unhindered and drop sprint tornadoes all over the place.
The gameplay is very fast paced. You'll be permanently sprinted, and depending on your variant, you may also have permanent wrath of the berserker. Your runspeed is very high, particularly in the latter case - and it has to be. You need to run from pack to pack before your fury drops, and before your wrath drops. This makes the build quite fun and active to play.
The "standard" build is essentially the following:
There is some room for adjustment, but not a whole lot. Lets go through the skills and explore the reasons for choosing them.
Battle Rage - Into the Fray: 15% damage and 3% crit are nice, of course, but the reason to take this skill is the chance to generate 15 fury on critical strikes. Paired with a good proc skill (Run Like the Wind) this is what is going to keep us swimming in fury.
Sprint - Run Like the Wind: This skill drives the build. Absolutely not optional. When we run around, we drop little tornadoes all over the place that deal AoE damage. They tic incredibly fast, do decent damage, and have a 20% proc coefficient for Life on Hit and Into the Fray. 20% may not seem like a lot, but with the speed they tic at, and the number you can get down, it is crazy good.
Whirlwind - Hurricane: At first blush a lot of people think this is our primary damage dealer, but it isn't - sprint is. Hurricane is our mobility skill. It lets us travel through enemies, dropping our sprint tornadoes all around and on top of them. It adds some damage, which is nice, but that is not its main function. It is important to think about it in these terms, so that you know to prioritize sprint when you're starved for fury, or forced to kite.
The three skills above make up the core of the build. Add to that the two skills that virtually every barbarian uses:
War Cry - Impunity: Need I say anything? Mountains of extra mitigation. Possibly the most overpowered defensive skill in the game. Certainly much better than any monk mantra. Virtually every barb build uses this skill, and for good reason. The only exceptions being builds for farming content you vastly over-gear.
Wrath of the Berserker: There are two possible rune choices for this build, each with some merit. The first, and my favorite, is Thrive on Chaos. We are going to be generating metric butt-loads of fury, so we can carry Thrive on Chaos across an entire zone, keeping the extra run speed, attack speed, and most importantly the crowd control immunity, the whole time.
The down side to Thrive is that it doesn't increase your damage all that much. If you want a 2 minute I-win button, rather than a near-permanent but far less powerful buff, Slaughter is your rune. With the kind of crit chance (and sheer number of crits thanks to Run Like the Wind) we have with this build, the Slaughter proc triggers dozens, maybe even hundreds of times over the course of a single Wrath activation. This rune will simply murder elite packs in no time. A pack you might otherwise struggle with is simply dead. Shielding packs are the only thing that can even hope to live through this, but their bubbles don't last as long as wrath. Of course if you go this route, you are going to run slower than a Thrive barb, and if you encounter a crowd-control heavy pack while wrath is on cooldown, you're kind of boned.
I recommend Slaughter if you are undergeared (particularly defensively undergeared), so that you can simply brute force an elite pack every 2 minutes. I recommend Thrive if you have decent (not necessarily great) gear, and you want to be killing elite packs more than once every 2 minutes. Thrive is absolutely the better rune for farming, as it gets you from pack to pack quicker.
All of this leaves us with one skill slot open. This is the slot where you'll see the most variation based on personal preference between different whirly barbs. I've chosen:
Bash - Instigation: Incredible fury generation, along the lines of the rest of the build. This is our only spammable "free" fury generator. Sprint generates sick amounts of fury, but it falls into the "it takes money to make money" category. If you run out of fury you can find yourself pretty boned. With instigation you take a whack or two and you're back laughing. You can also bash destructible objects to get fury for sprints (and help keep thrive up) while traveling between packs.
Other common choices include:
Overpower - Crushing Advance: I'm nerfed, don't use me. Old content spoilered if you're interested.
SpoilerOverpower - Crushing Advance: This is simply an incredible defensive (and offensive) tool. With the amount of crits this build spams out, the cooldown on Overpower will be up very very quickly, allowing you to practically have the Crushing Advance buff be permanent. With decent Life on Hit Crushing Advance alone can heal you from standing in desecrate or molten. This skill will dramatically improve your survivability if you play perfectly. So, why wouldn't everybody take it? It doesn't generate fury. If you take Overpower, your only way to get fury when you're "cold" is war cry - and of course, taking hits. Overpower means you'll almost never die in high fury situations, but it greatly increases the danger of low fury situations. If battle rage drops, or if you're fighting an annoying ranged elite that keeps running out of your tornadoes, it is easy to bone yourself. If you can't refresh battle rage and sprint, you are humped.
If you run Overpower instead of a fury generator, it is highly recommended that you use a mighty weapon in your mainhand, so that sprints can generate fury without battle rage up. War cry generates enough fury for either sprint or battle rage, but not both. With a mighty weapon you can simply always elect to use sprint, and then hit battle rage when you get enough fury.
Virtually any other generator is an okay choice here too. Some people use Iron Impact, because wallers are a freaking pain for this build. Some people use Frenzy - Maniac, to get better single-target DPS for popping loot goblins, single ranged enemies, and that sort of thing. There's other valid choices too.
Ruthless: Our build is crit dependent, so obviously more crit chance and crit damage is great. This is pretty much mandatory.
Weapon Master: 10% crit (mace or axe) or 3 fury per hit (mighty weapon). Pretty much mandatory. Regardless of whether you use a mace/axe or mighty weapon, the function is essentially the same - more fury generation. Also more damage in the case of mace/axe.
The third passive is a flex slot, you can put whatever you like. Some common options:
Nerves of Steel / Tough as Nails: Self explanatory. Use whichever gives you more armor with your gear. Obviously choose one of these if you feel a little squishy in the content you're doing.
Superstition: In general I favor armor bonuses, as they effect all damage, however if you primarily find yourself having trouble with things like arcane sentries, superstition is worth considering. You won't get 20% damage reduction from one of the armor passives, that's for sure. The exact amount you get varies, but for me personally tough as nails gives 12.29% damage reduction, so that should give you a ballpark for comparison.
Animosity: More fury! We love fury! This will make carrying thrive around easier, so long as you manage to spend it fast enough. The bigger pool is also surprisingly nice, as you can spend a pool pretty fast spamming whirl, battle rage, and sprint, only to realize that stuff has moved out of your tornadoes or get a string of no-crits. Bigger pool is good.
Unforgiving: I tend to prefer animosity, but having fury regenerate instead of degenerating can take a load off your mind in terms of the pace of play. Also a decent choice if you're not using a fury generator in your active skill flex slot. The fury generation is trivial in-combat, but being able to choose to start any fight at full fury is a pretty nice advantage (or kite until you can get battle rage and sprint back up if you're not using a generator).
Mainhand: Mace, axe, or mighty weapon. Mace is generally the best choice. Mighty weapon is an interesting choice, particularly if you aren't using a fury generator of any kind, as you can generate fury from overpower. However, you're going to do the most damage with a mace. Run Like the Wind's tornadoes always deal your mainhand damage, but they will do it at the speed of your current weapon (which alternates mainhand, offhand). As such, a slow mainhand and fast offhand optimizes our damage from Run Like the Wind - which is our primary damage source. That said, an axe isn't much faster than a mace, so if you find a good one, by all means, use it. Good stats to have are crit damage and life on hit. A socket is a good way to get a bunch of extra crit damage. Strength of course is always welcome.
Offhand: As mentioned above, Run Like the Wind deals mainhand damage, but sometimes it does it at offhand speed. This gives us two priorities for our offhand: speed and stats - not damage. Since our primary damage dealer doesn't care at all about our offhand damage, we can save money by getting one with low DPS. Ideally we want a dagger because they are fast. If it has attack speed that's great too. Beyond that, life on hit and crit damage are the stats to look for. A socket is a good way to get a bunch of extra crit damage. Strength of course is always welcome.
Armor stat priorities:
Critical Hit Chance: Our goal is to have ~50% crit with a mace or axe, or 40% crit with a mighty weapon - you can get away with less, but that's a good target. Crits are our life. For some builds crit is just a way to do extra damage, but for us they are our fuel. Stack the heck out of crit.
Strength: Damage and armor, what's not to like?
Vitality: Life and armor (if you use Nerves of Steel). I recommend ~40k health for a3/a4 - I'm a little shy of this at the moment myself, and really feeling it.
Resist All / Armor: These stats do essentially the same thing. For us, because we have a huge resist buff from Impunity, and because we have naturally high armor, resists are generally better. However, when considering two pieces, one with bonus armor, the other with bonus resist all, it is best to simply use an Effective Health Calculator and pick whichever gives the most EHP.
To give you a ballpark on what is good. I'd recommend 400 res all and 6k armor for act 1, 600 res all and 7k armor for act 2, and 800 res all and 8k armor for act 3/4. Less armor and more resists, or vice versa, is fine. These are just ballpark numbers.
Life on Hit: Run Like the Wind's crazy procing isn't just good for fury generation, it is also good for crazy LoH returns. While the best place to get LoH is weapons, if you can get it on your jewelry, it is great to have. I don't have any exact value to recommend to you - more resists/armor means less LoH is needed (so does just killing stuff super fast), but IMHO 1000 is a bare minimum of LoH for acts 3/4, and I'd prefer 1500 or more.
Attack Speed: More damage, more life on hit procs, more fury because of more crits. What's not to like? Assuming you're in the neighborhood of 50% crit, attack speed has roughly half the value of crit chance in terms of fury generation. However due to the increased LoH healing and damage, its value is a bit higher than that. It is a judgement call exactly how heavily to weight this stat, but it is definitely good.
Critical Hit Damage: A great damage stat, particularly on weapons which can get a lot of it, but it is also available on gloves and jewelry. I tend to favor attack speed and critical chance if possible on these slots, but if you can get crit damage, great. There's no hidden extra value to crit damage like there is with crit chance and attack speed, just value it based on the amount of damage it adds on your char screen.
Life Leech: Life leech scales with damage done, so whether it is good for you depends on how much total damage you are outputting. For us mere mortals with 20-50k character screen damage, life leech is firmly in the "not worth it" category. If you happen to get it with other stats, great, but don't go looking for it. If you're at the ultra-high end of gear life leech is worth considering. Even with 100k chracter screen damage you'll get more life from 800+ LoH on a weapon than from 3% life leech, however, you can get life leech on a mighty belt - a slot that cannot get LoH - so it is worth picking up there. It is also possible for a weapon to spawn with both LoH and LL on it.
If you use Slaughter instead of Thrive on Chaos, life leech can give substantial returns even with modest gear, during the slaughter. If you're following the two-min-win style, it is therefor reasonable (and potentially cheaper) to consider LL over LoH. Your returns will be pitiful most of the time, but when you really need it is when you're slaughtering a tough pack, and that's when it will pay off.
So you're geared out and spec'd out. Now you need to figure out how to play the bloody character. It is on the higher end, complexity wise, of Diablo 3 builds (not that that's saying a whole lot). There are two things that should be on your mind while playing. The first is monitoring your buffs. We have lots of buffs, and we need to keep all of them active, or we fall over and die. Impunity needs to be up, obviously, but then we've got battle rage, sprint, and thrive on chaos to keep active.
Battle Rage is the buff that messes with me the most. Its duration is too long for it to constantly be in my mind, but too short to just cast it once per fight - but it is absolutely essential to keep it up. In high fury fights I simply spam it so I won't forget it (and to keep my fury down for Thrive on Chaos).
Sprint needs to be up all the time, pretty much. The worst thing you can do is start whirlwinding without sprint active. You'll just tank your fury without doing much damage or healing much from LoH. Fortunately it is very very easy to feel when sprint drops, as your character will slow to a crawl.
Thrive on Chaos is the most active and interesting buff to monitor. You refresh the duration by gaining fury, but only if your fury bulb isn't already full. As such you often need to spam away your fury on sprint and battle rage in order to make room for new fury to be generated, adding time to Thrive on Chaos.
One annoying wrinkle is that the buff bar only shows 5 buffs total. Valor + War Cry + Battle Rage + Sprint + Thrive = 5 buffs, so if you have any kind of other buff, such as a shrine, a monk aura, etc, your most recently cast buff won't show up on the buff bar. This can be particularly annoying when that buff is Thrive on Chaos, as you want to be able to monitor how much time you have left, so you know if you can make a run for the next pack or if you should stop to bash some stuff or whatnot. Sometimes the solution is to simply click off the shrine buff, as much as that hurts to do. Otherwise you're just flying on faith and memory.
I said there were two things that needed to be in your mind. We've talked about buffs, the other is positional awareness. With whirlwind, sprint, and thrive on chaos we have unprecedented mobility. If we're aware of what's going on on the screen, ground effects should almost never get us. We can whirl around and between arcane beams quite easily, so long as we're paying attention.
It starts to get tricky when you add walls in. Waller is our number one worst affix, no question. We don't have leap in our build, so getting boxed in is death. Whenever possible, we want to pull wallers into as open of an area as we have available to us, so that we can whirl our way around walls and not get trapped.
If we're forced to fight indoors, in tight hallways, there's not a lot we can do about wallers other than hope things go well. Of course the walls by themselves aren't so bad, but when they force you to stand in desecrate or arcane, while preventing you from getting any LoH healing from sprint, it is a problem. This is actually the situation where Crushing Advance shines. If you get walled in and desecrated, Crushing Advance will generally keep you alive - unless the enemies you're fighting are ranged, then you're probably still humped. Walled into desecrate/arcane is pop a potion and pray time.
I could say more about play style, but it is easier to show than tell, so:
Here's a demonstration video of my modestly geared whirly barb farming some elites in Act 3. My gear has a total value of about 10 million gold, so don't expect to be impressed by stuff falling over dead like you might see in a Krip video. What you will see is how the build plays (and me failing to refresh battle rage at one or two points).
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Any comments, corrections, or additions are welcome. I'm not really an expert (only 40 hours played on my barb), but I thought we needed an informational thread about what has become the most popular barbarian build.